Seismic Data Filtering Using Non-Local Means Algorithm Based on Structure Tensor

Open Geosci. 2017; 9:151–160

Research Article Open Access

Shuai Yang, Anqing Chen*, and Hongde Chen Seismic data filtering using non-local means algorithm based on structure tensor

DOI 10.1515/geo-2017-0013 Received Sep 22, 2016; accepted Feb 03, 2017 1 Introduction

Abstract: Non-Local means algorithm is a new and ef- Filtering is a key step in seismic data processing [1]. Due to fective filtering method. It calculates weights of all sim- increasing demands for high-quality data, interpretation- ilar neighborhoods’ center points relative to filtering oriented processing becomes more and more important. point within searching range by Gaussian weighted Eu- Improved signal-to-noise ratio (S/N) by filtering process- clidean distance between neighborhoods, then gets fil- ing may facilitate subsequent studies of seismic geomor- tering point’s value by weighted average to complete phology, seismic facies and micro-structures as well as the filtering operation. In this paper, geometric distance reservoir prediction. of neighborhood’s center point is taken into account in Seismic data filtering may be realized through Gaus- the distance measure calculation, making the non-local sian filtering, median filtering, mean filtering and vari- means algorithm more reasonable. Furthermore, in order ous transform-domain approaches, e.g. Fourier transform- to better protect the geometry structure information of based f -k and f-x filtering, wavelet transform-based filter- seismic data, we introduce structure tensor that can de- ing [2, 3], curvelet transform-based filtering [4] and Radon pict the local geometrical features of seismic data. The co- transform-based filtering. Additionally, there is the filter- herence measure, which reflects image local contrast,is ing technique based on a geostatistical method which extracted from the structure tensor, is integrated into the makes use of the variogram function after decomposition non-local means algorithm to participate in the weight to do geostatistical estimation respectively to get the es- calculation, the control factor of geometry structure sim- timated value of the various components of the original ilarity is added to form a non-local means filtering algo- data, including the effective information and noise values. rithm based on structure tensor. The experimental results It has been applied to filtering of seismic data [5]. These prove that the algorithm can effectively restrain noise, with techniques are effective for seismic data filtering, but most strong anti-noise and amplitude preservation effect, im- of them have the disadvantages of blurring the details proving PSNR and protecting structure information of seis- and edges of an image in the process of noise reduction. mic image. The method has been successfully applied in Marginal reflections or seismic discontinuities may be seismic data processing, indicating that it is a new and ef- caused by faults, fractures, channels, lenticular geobodies fective technique to conduct the structure-preserved filter- or reefs. Such features are closely related to the interpre- ing of seismic data. tation of hydrocarbon reservoirs. Consequently some new techniques, e.g. filtering based on mathematical morphol- Keywords: Non-Local means; Structure tensor; Filter; Sim- ogy [6], structure constrained edge-preserved filtering [7], ilarity; Interpretative processing and anisotropic diffusion filtering based on partial differ- ential equation [8], have been developed to preserve edge structures or the contacts between reflections and forma- Shuai Yang: College of Geosciences, China University of tions. Petroleum(Beijing), Beijing, 102249, China; Institute of Sedimen- Buades et al. made a comparative study of some typi- tary Geology, Chengdu University of Technology, Chengdu, 610059, cal filtering techniques and developed a non-local means China; Email: [email protected] (NLM) algorithm for noise removal [9–11]. This algorithm *Corresponding Author: Anqing Chen: Institute of Sedimen- tary Geology, Chengdu University of Technology, Chengdu, 610059, was demonstrated to have better performance than other China; Email: [email protected]; Tel.: +86-028-84073252; typical denoising methods. David Bonar et al. success- Fax: +86-028-84078992 fully applied the NLM algorithm in seismic data denoising Hongde Chen: Institute of Sedimentary Geology, Chengdu with the idea of neighborhood matching and correlation University of Technology, Chengdu, 610059, China; Email: as well as weighted smoothing [12, 13], but they did not [email protected]

take account of seismic geometric properties such as di- and its neighborhood Ni to the pixel j and its neighborhood rectivity and local contrast. Apparently, if a seismic pro- Nj. Note that each pixel i has its own independent weight- file or attribute slice is considered as a digital image, each ing factor of the other pixels j within the image, which is pixel would be characterized by two properties, i.e. numer- calculated by Equation (3). ical property of the pixel itself and geometric structure of (︂ )︂ 1 −D2(i, j) neighboring pixels. In this paper, we develop a new filter- w(i, j) = exp (3) Z(i) h2 ing algorithm for seismic data (image) processing. It inte- ∑︀ grates similarity evaluation by local contrast with NLM fil- where Z(i) is the normalizing factor to ensure w(i, j) = 1, tering based on structure tensor, which captures the geo- j and defined by metric structure of an image [8]. The calculation of neigh- (︂ )︂ borhood similarity distance involves numerical similar- ∑︁ −D2(i, j) Z(i) = exp (4) h2 ity and neighborhood geometric distance. A coherence at- j tribute, designed to describe local contrast, is added into weighting factor estimation to evaluate similarity. A con- The filtering parameter h is a constant which controls trol factor is also included to adjust the weight of geomet- the decay rate of the exponential function and thus deter- ric property. As per model data processing and residual er- mines the degree of filtering. For example, a large value ror images, this algorithm achieves a better result of edge for h will provide very similar weight for all pixels j, thus preservation. In addition, a large control factor reduces the the image would be blurred. On the contrary, a small value weight of geometric property for better noise attenuation. for h will provide a significant weight for only a few ofthe The results of field data processing also show that this al- pixels j, thus noises cannot be attenuated sufficiently. 2 gorithm is capable of edge preservation while noise sup- The Gaussian weighted Euclidean distance D (i, j) in pression. Equation (3) is defined by the following expression. D2 i j ⃒v N v N ⃒2 ( , ) = ⃒ ( i) − ( j)⃒2,a (5) nl ∑︁ [︀ ]︀2 2 Theoretical studies = Ga(xl , yl) (v(Ni(l)) − v(Nj(l))) l

2.1 NLM filtering | |2 where the operator • 2, a denotes the squared factor of 2 Gaussian weighted Euclidean distance D (i, j), Ni repre- Previous denoising algorithms were developed under the sents a neighborhood with the center at the pixel i, which assumption that the raw image is regular. Moreover many is usually a square domain, Ga represents the Gaussian details of the image may also be smoothed in denoising. kernel with the standard deviation a, and l represents one These two disadvantages have been remedied by NLM fil- of the total nl elements within a neighborhood. tering developed by Buades et al. [14]. The basic idea of For a 2D image, the Gaussian kernel can be defined by, this algorithm is that an image usually contains a mass of (︂ 2 2 )︂ noises, thus an image v is defined as (x − x0) + (y − y0) Ga(x, y) = exp − (6) 2a2 v = u + n (1) where X0 and y0 represent the center of Gaussian kernel As per Equation (1), the image v is composed of the with x and y corresponding to the coordinates of the ele- original noise-free image u and random noise n. At the ment l in equation (5). pixel i, the NLM filtering result ̂︀v(i) is simply the weighted For three neighborhoods (in red, green and blue, re- average of all of pixels within the noisy image v. spectively) around a square image (in Figure 1a), the ∑︁ weights of the centers in respect to other all of elements ^v(i) = w(i, j)v(j) (2) within the image are calculated by Equation (3). We ob- j viously can see, for red neighborhood (in Figure 1a), the In Equation (2), the weighting factor w(i, j) is dependent centers of all red neighborhoods in Figure 1b have larger on the similarity between pixels i and j, in addition must weighting factors due to good similarity. Similarly for ∑︀ satisfy the conditions 0 ≤ w(i, j) ≤ 1 and w(i, j) = 1. The green neighborhood (in Figure 1a), the centers of all green j neighborhoods in Figure 1c have larger weighting factors. similarity between the pixels i and j is represented by Gaus- For blue neighborhood (in Figure 1a), the centers of all blue sian weighted Euclidean distance D(i, j) from the pixel i neighborhoods in Figure 1d have larger weighting factors. Seismic data filtering using non-local means algorithm Ë 153

center may be somewhat mitigated. In numerical calculation, w(i, i)=0.5 or w(i, i) = max(w(i, j)∀i≠j). In following calculation, w(i, i) = 0.5. Each pixel in the image would be compared with all pixels in the process of NLM filtering, which results in mass computation and low efficiency. For an image with M×N pixels, M×N weighting factors would be calculated repeat- edly for each pixel. Therefore altogether (M×N)2 factors would be calculated in the filtering. It is difficult to pop- (a) (b) ularize such an inefficient algorithm. More importantly, many irrelevant neighborhoods with large distance are assigned weighting factors, this would impair the result of filtering. Buades [14] proposed that the region of search is lim- ited within a square area, i.e. a neighborhood, around the pixel to be handled so as to improve efficiency of filtering. Thus for a region of S×S and an image with M×N pixels, the computational complexity is M×N×(S×S−1) (because it is unnecessary to calculate the weighting factor for the (c) (d) center of the neighborhood itself). As a result, the computational complexity is greatly reduced compared with the Figure 1: The weights schematic diagrams of Non-Local means algo- original NLM algorithm. rithm

As shown in the Figure 1 and Equation (5), for two 2.2 Structure tensor and local contrast neighborhoods with identical numerical distribution and structures, the square factor of Gaussian weighted Eu- The definition of structure tensor proposed by J. Weickert clidean distance between them is equal to zero. Therefore, et al. [8] is very useful in image analysis to estimate the for the center of red neighborhood (in Figure 1a), the cen- direction field and extract local structures of the image. ters of all red neighborhoods in Figures 1b have identi- T Sρ = Gρ * (∇vσ · ∇vσ ) (8) cal weighting factors, which is obviously unreasonable. ⎡ ⎤ 2 2 (︁ ∂v )︁2 (︁ ∂v ∂v )︁ Hence, D (i, j) in Equation (3) is corrected to be D (i, j) by σ * Gρ σ σ * Gρ ⎢ ∂x ∂x ∂y ⎥ the idea of inverse distance weighted interpolation. = (︁ )︁ (︁ )︁2 ⎣ ∂vσ ∂vσ ∂vσ ⎦ ∂x ∂y * Gρ ∂y * Gρ 2 D (i, j) = D2(i, j) + |i − j| 2 (7) where Sρ is the structure tensor, ∇v is the gradient of the where |i−j| is the geometric distance between the pix- image, vσ=v * Gσ is the image of v after Gaussian filter- els i and j, for the pixel i with the coordinates of xi ing, Gσ is the Gaussian kernel defined by σ. A procedure of and yi and the pixel j with the coordinates of xj and yj, Gaussian filtering before gradient estimation could elimi- 2 2 2 |i−j| =(xi−xj) + (yi − yj) . nate the impacts of noises, ρ is the variance of the Gaus- In addition, there is a defect in neighborhood distance sian filter Gρ. More information involved by the convolu- measurement in the Equation (5). The center of the neigh- tion of Gρ may facilitate the delineation of such geometric borhood has a much larger weighting factor than other structures as linear laminae, fracture boundaries and an- all of elements within the region of search, which means gular regions. if i=j, the contribution from adjacent elements to simi- The structure tensor is a symmetric positive semi- larity is reduced by over-weighting. For example, if the definite matrix as per the definition. In accordance with center X is a noisy pixel and surrounding pixels are less the principles of matrix eigenvector decomposition, the noisy, the distance calculated by Equation (5) with the structure tensor is decomposed to be large weighting factor for X is unfavorable for denoising. [︃ ]︃ [︃ ]︃ [︃ ]︃ [︁ ]︁ T w i i s11 s12 λ1 0 υ1 In order to solve this problem, we adopt that 0≤ ( , )<1 Sρ = = υ υ · · (9) s s 1 2 λ υT when i=j. Then, the impact of the weighting factor at the 12 22 0 2 2 154 Ë S. Yang et al.

ence parameter for quantitative description of local contrast.

2 2 2 H=(λ1−λ1) =(s11−s12) +4s12 (12)

H indicates local contrast of pixel.

2.3 NLM filtering based on structure tensor

A seismic profile or attribute slice has two properties, numerical property of the pixel itself and geometric structure of neighboring pixels. Large local contrast usually occurs at the marginal structures and faults, while small local contrast occurs in the flat regions. Hence, it is inade- Figure 2: The schematic diagram of elliptic model about structure quate to involve only numerical similarity in weighting fac- tensor tor calculation for neighborhood similarity. According to the definition, structure tensor describes local geometric properties of the image. Therefore, the coherence parame- For two eigenvectors solved, assume λ1 > λ2, i.e. a positive ter H which describes local contrast should be considered λ1 and negative λ2. in weighting factor calculation. (︂ √︁ )︂ 1 2 2 In accordance with the idea of bilateral filter [15], the λ1 = s11 + s22 + (s11 − s22) + 4s (10) 2 12 weighting factor w(i, j) for NLM filtering algorithm based (︂ )︂ 1 √︁ on structure tensor is defined as follows. λ = s + s − (s − s )2 + 4s2 2 2 11 22 11 22 12 (︂ )︂ (︂ )︂ 1 −D2(i, j) −H2(i, j) w(i, j) = exp · exp (13) Z(i) h2 h2 Two corresponding unit orthogonal eigenvectors υ1 and υ2 are The normalized factor is ⎧ (︃ )︃ ∑︁ (︂ D2 i j )︂ (︂ H2 i j )︂ ⎪ 2S12 − ( , ) − ( , ) ⎪ω = √︁ Z(i) = exp · exp (14) ⎪ S S S S 2 S 2 h2 h2 ⎨ 22− 11+ ( 22− 11) +4 12 j ω (11) ⎪υ1 = |ω| ⎪ Similar to D2(i, j) in Equation (5), H2(i, j) is defined by the ⎩⎪υ υ⊥ 2 = 2 following expression.

H2 i j ⃒H N H N ⃒2 In the local coordinate system composed of two eigen- ( , ) = ⃒ ( i) − ( j)⃒2,a (15) vectors at (x, y), υ1 represents the direction in parallel to nl ∑︁ [︀ ]︀2 ∇v with the maximum rate of change or the direction with = Ga(l) (H(Ni(l)) − H(Nj(l))) the maximum contrast in the geometric structure, which l corresponds to the vertical direction of the structure. υ2 In practical application, a geometric similarity control stands for the direction perpendicular to ∇v with the min- factor δ [16] is designed in weighting factor calculation to imum rate of change or the direction with the minimum adjust H2(i, j), then w(i, j) is defined to be contrast, which corresponds to the direction of the struc- (︂ )︂ (︂ )︂ 1 −D2(i, j) −δH2(i, j) ture. There are three scenarios on seismic images. w(i, j) = exp · exp (16) Z(i) h2 h2 1. λ1≈λ2≈0 indicates blank reflections or similar reflec- δ∈ δ tion strength in a fat region. Note that [0, 1]. When the is equal to 0, the algorithm takes the original form of NLM filtering. An expo- 2. λ1 >> λ2≈0 indicates marginal reflections caused by ∈ faults, pinch-outs, etc. nential factor (0, 1] is added compared with a NLM algorithm. As per Equation (15), the exponential factor is equal 3. λ1≥λ2 >> 0 indicates reflection termination in angu- H i.e. lar regions. to 1 when two neighborhoods have identical , identical geometric structures (H2(i, j)=0), when weighting fac- Local features of each pixel in the image are described tors are mainly dependent on the numerical similarity of by the variations in two directions. Here we define a coher- the neighborhoods. Seismic data filtering using non-local means algorithm Ë 155

A seismic data volume differs greatly from a conven- 8. In terms of the filtering parameter h and control fac- tional image in data range. The order of amplitude varia- tor δ of geometric similarity, use Equation (16) to cal- tions of seismic data processed by different operators may culate weighting factor w(i, j) of all neighborhood reach n-th power of 10. Consequently, the coherence pa- centers within the region of search in respect to the rameter H also has large data range. The filtering param- element to be evaluated. eter h and control factor δ should be adjusted as per the 9. Use Equation (2) to calculate the weighted average of scale and coherence parameter of seismic data. The con- all neighborhood centers within the region of search trol factor δ is usually very small to offset the impacts of to obtain the value of the element to be evaluated. large H2(i, j). The factor δ adjusts the contribution of geo- 10. Fulfill the filtering of all elements. If the result isnot metric properties to weighting factors to generate different acceptable, return to step (1) to adjust parameters for results of filtering. An increase in δ makes the exponential recalculation until the result is satisfactory. factor, as well as weighting factors decrease. Such process In this workflow, the coherence parameter H of each may preserve geometric properties but cannot eliminate element is calculated first because the structure tensor is noises sufficiently. This parameter should be adjusted as only related to data distribution in its neighborhood. per the requirements of data processing, which would be discussed in the following model tests. According to Equation (16), neighborhood similarity involves numerical similarity and geometric similarity. If 3 Case studies such a neighborhood is searched to match the original neighborhood, it would be assigned with a large weight- In order to evaluate the results of NLM filtering based on ing factor. Thus geometric properties are better preserved structure tensor, a layered model was designed with faults after processing. and wedge-like pinch-outs (Figure 3). Interval velocities The steps are detailed as follows. were also defined. The underground structure was gridded to be a discrete model with 128 traces and 200 sampling 1. Define the size of S×S, e.g. 21×21, for a region of points per trace. The synthetic seismic profile (Figure 4) search and the neighborhood size (for example, 3×3 was made with a Ricker wavelet designed with the domi- or 5×5). Input a proper Gaussian function standard nant frequency of 30 Hz, 64 sampling points and sampling deviation a, filtering parameter h and control factor interval of 2 ms. Figure 5 shows the record with 10% ran- δ. dom noises. Figure 6 shows the coherence calculated from 2. For boundary data filtering, use mirror reflections to the noisy model data (Figure 5). The coherence attribute extend the original boundary of an image as per the is reconciled with the geometric properties of model data size of neighborhood. For example, if the neighbor- and thus could be used to describe local contrast. hood is of 5×5, the boundary would be extended with Figure 7 shows the result after Gaussian filtering of 2 data. noisy model data (Figure 5). The neighborhood size is 3. Conduct Gaussian smoothing for all elements in the image with the operator G(0, σ2) in accordance with neighborhood size. 4. Use the Sobel operator or central difference to calculate partial derivative. (︁ )︁2 2 ∂vσ 5. Convolve the Gaussian filter G(0, ρ ) and ∂x , (︁ )︁2 ∂vσ ∂vσ ∂vσ ∂y and ∂x · ∂y , respectively to obtain three elements S11, S12 and S22 of the structure tensor matrix. 6. Calculate coherence parameter H as per Equation (12) to describe geometric similarity. 7. For any element to be evaluated in the image, calcu- 2 late the distance D (i, j) between the neighborhood of the element to be evaluated and that of match element and local contrast H2(i, j) as per neighborhood Figure 3: Multi-layer geological model contained fault and wedge size. pinch out 156 Ë S. Yang et al.

Figure 4: Synthetic seismic record Figure 7: Gauss filtering’s result

Figure 5: Seismic record of added 10% random noise Figure 8: Noise of Gauss filtering

Figure 6: Result of coherence measure Figure 9: Filtering result of NLM algorithm

−12 11×11, and variance is 1. Figure 9 shows the result of NLM rameter h=50 and δ = 2.5 × 10 . Figures 8, 10 and 12 filtering with the region of search 21×21, neighborhood show noise images derived from three algorithms respec- 11×11, Gaussian weighted variance 1 and filtering parame- tively. Gaussian filtering has eliminated noises but also ter h=50. Figure 11 shows the result of NLM filtering based over smoothed events and faults and pinch-out reflec- on structure tensor with the region of search 21×21, neigh- tions, so this algorithm is not an ideal method for seis- borhood 11×11, Gaussian weighted variance 1, filtering pa- mic data filtering. NLM filtering and NLM filtering based on structure tensor have generated very similar results (Fig- Seismic data filtering using non-local means algorithm Ë 157

ure 11) has a higher peak S/N of 59.2443 compared with the value of 57.3538 for the former (Figure 9). In conclusion, in comparison with NLM filtering, the algorithm of NLM filtering based on structure tensor has better performance of noise suppression and edge preservation. In field data processing by NLM filtering based on structure tensor (Figure 13), the seismic profile has 551 traces and 1000 sampling points per trace. The sampling interval is 2 ms. Processing parameters are: (1) the region of search 11×11, neighborhood 5×5; (2) Gaussian weighted variance 1; (3) filtering parameter h=1000 and δ = 10−18. The results of filtering and noise image are shown in Fig- ures 14 and 15, respectively. It can be seen that after fil- Noise of NLM algorithm Figure 10: tering the noise is depressed effectively, while the texture of seismic profile is well preserved, especially fault reflections have been improved significantly, with clearer fault boundaries and fault combination features (white cir- cle area in Figure 13 and Figure 14), and event continu- ity has been enhanced. The part (white rectangular area in Figure 13) that can be interpreted falsely as a weak- continuous reflection in the noise profile, shows the fea- ture of lens shape after filtering, which is closer to the geometry structure of the geological body (white rectangular area in Figure 14). It is favorable for the identification of river and beach bars. Figure 16 shows a seismic amplitude slice with two suspect channels, in which a large meandering river chan- Figure 11: Filtering result of NLM algorithm based on structure nel is shown on the left of the center and a small mean- tensor(δ = 2.5 × 10−12) dering river channel is shown in the lower left, and this figure is contaminated by some noises, which is not good for a study on characteristics of seismic topography. Af- ter NLM filtering based on structure tensor with the region of search 7×7, a neighborhood of 5×5, the Gaussian weighted variance 1, filtering parameter of h=1000, and δ = 10−9. The result (Figure 17) and noise image (Fig- ure 18) show improved S/N and enhanced channel boundaries and structures. Such result provides more evidence for further studying the geomorphic features and seismic facies. Buades [10] demonstrated that NLM filtering with neighborhood size of 5×5, 7×7, 9×9 and 11×11 had a good anti-noise capacity and also preserved the details or geometric properties of the original image. For seismic data fil- Figure 12: Noise of NLM algorithm based on structure tensor(δ = tering, the filtering parameters should be selected accord- 2.5 × 10−12) ing to the range of amplitude variations due to the very large amplitude variations of seismic data. ure 9 and Figure 11). The former has eliminated noises effectively (Figure 10), however, the latter has also effectively protected events and fault and pinch-out reflections with apparent geometric properties in addition to denoising (Figure 12). Furthermore, the result of the latter (Fig- 158 Ë S. Yang et al.

Figure 13: Practical seismic profile contained noise

Figure 14: Filtering result of seismic profile

Figure 15: Noise of NLM algorithm based on structure tensor Seismic data filtering using non-local means algorithm Ë 159

4 Conclusions

The calculation of neighborhood similarity distance involves numerical similarity and neighborhood geometric distance. If some neighborhoods are searched to match the original neighborhood, the one with the smallest geometric distance would be assigned with the largest weighting factor. The coherence attribute, designed to describe local contrast, is adopted to enhance the sensitivity of distance to noises in the region with small contrast. Hence, the weighting factor could be adjusted by NLM denoising algorithm for the center of each similar neighborhood in a noisy region with similar reflection strength. Our test of

Figure 16: Seismic amplitude slice contained noise model data prove that the algorithm could preserve the structures of the image and generate high peak S/N, especially for the seismic data with significant geometric properties. In addition, this algorithm has been demonstrated practically and credibly in processing field data.

Acknowledgement: The authors are grateful to the anony- mous reviewers whose suggestions have been helpful in preparation of the manuscript, and thank editor for care- fully revising the manuscript.

References

[1] Yilmaz, Ö., Seismic data analysis: Processing, inversion, and interpretation of seismic data, 2nd ed.: SEG., 2001 [2] Li Z.C., Zhang Y.C., Wang Q.Z., Joint denosing method based on Figure 17: Filtering result of seismic amplitude slice wavelet transform and multi-level median filtering. Geophysi- cal Prospecting for petroleum, 2009, 48(5), 470-474 (in Chinese with English abstract) [3] Chen X.P., Cao S.Y., Second-generation wavelet transform and its application in the de-noising of seismic data. Geophysical Prospecting for petroleum, 2004, 43(6), 547-550 (in Chinese with English abstract) [4] Peng C., Chang Z., Zhu S.J., Noise elimination method based on curvelet transform. Geophysical Prospecting for petroleum, 2008, 47(5), 461-464(in Chinese with English abstract) [5] Henning H., Thierry C., David L.M., Erika A., Pierre L. and Didier L., On the use of Geostatistical Filetring Techniques in seismic processing. 73rd SEG International Exposition & Annual Meet- ing, Dallas, 2003, 7-11 [6] Serra J., Image Analysis and Mathematical Morphology. Aca- demic Press, New York, 1982 [7] Wang J., Chen Y.H., Xu D.H., Qiao Y.L., Structure-oriented edge- preserving smoothing based on accurate estimation of orienta- tion and edges. Applied Geophysics, 2009, 6(eq4), 367-376 Figure 18: Noise of NLM algorithm based on structure tensor [8] Weichert J., Anisotropic diffusion in image processing. PhD thesis, University of Kaiserslautern, Germany, 1996 [9] Buades A., Coll B., Morel J.M., A review of image denoising algorithms, with a new one. Multiscale Modeling and Simulation, 2005, 4, 490-530 160 Ë S. Yang et al.

[10] Buades A., Coll B., Morel J.M., Image denoising methods. a new [14] Buades A., Coll B., Morel J.M, Nonlocal image and movie denois- nonlocal principle. SIAM Review, 2010, 52, 113-147 ing. International Journal of Computer Vision, 2008, 76(2), 123- [11] Buades A., Chien A., Morel J.M., Osher S., Topology preserv- 139 ing linear filtering applied to medical imaging. SIAM Journal on [15] Tomasi C., Manduchi R., Bilateral filtering for gray and color im- Imaging Science, 2008, 1(1), 26-50 ages. Sixth International Conference on Computer Vision, Bom- [12] David B., Mauricio S., Denoising seismic data using the nonlocal bay, 1998, 839-846 means algorithm. Geophysics, 2012, 77(1), A5-A8 [16] Xu J., Research of image denoising using Non-Local Means al- [13] Wang W.H., Guo X.B., Shi Y.,Non-local means filtering denoising gorithm. Master thesis, Nanjing University of Science and Tech- approach and its application in VSP reverse time migration seis- nology, China, 2009(in Chinese) mic data. Geophysical Prospecting for petroleum, 2015, 54(2), 165-171(in Chinese with English abstract)